Physics 263: Problem Set #4

1. (BTM 2.1.4) Two ways you might compare the two integrals are i) graphically and ii) with a change of variables (e.g., x' = pi/2 + x).
2. (BTM 2.2.3) From problem 2.1.3 you have the integral definition of the gamma function, which converges absolutely for any argument with a positive real part. Gamma[z] is the integral from 0 to infinity of x^z-1 e^-x dx. Note also that Gamma[z+1] = z*Gamma[z] (like with factorials). These facts plus a basic substitution should be enough to take you to the desired answer.
3. (BTM 2.2.4) The first part is just an application of implicit differentiation (note that Delta y/Delta x is approximately dy/dx). After you substitute the expression for y in terms of x, be sure to put everything under the square root over a common denominator, which leads to a great simplification. Then it is a standard trig substitution.
4. (BTM 2.2.10) This is a nice problem! Just follow the instructions and it comes out directly. (I can't think of any additional hints!)